##  [Universal Cover](/universal-cover) 

  ##  [Universal Cover](https://puremath.quantumdictionary.io/universal-cover-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A covering space p: Ũ → X that is simply connected (Ũ is path-connected and π1(Ũ) is trivial) and covers X; when it exists, the universal cover maps onto the base and every other connected covering of X factors through it. It is unique up to homeomorphism over X.